论文标题
五角形几何形状具有块尺寸3、4和5
Pentagonal geometries with block sizes 3, 4 and 5
论文作者
论文摘要
五边形几何pent（$ k $，$ r $）是一个部分线性空间，每个线路或块都带有$ k $点的事件，每个点都与$ r $ lines发生事件，并且对于每个点$ x $，都有一个线路事件，这些点事件与$ x $ collinear noces noces culine五边形几何形状中的相对线对由两个平行线组成，使得一条线上的每个点与另一线上的那些点都不与这些点进行界线。 我们为具有块尺寸3和连接缺陷图的无限五边形几何形状序列提供了直接的结构。另外，我们介绍了39个新的五角形几何形状，具有块尺寸为4和5，块尺寸为5，均具有连接的缺陷图。因此，我们确定了不包含相反线对的pent（4，$ r $）的一些可能例外的存在频谱，以及带有一对相对行对的pent（4，$ r $）。更普遍地，考虑到$ j $，我们表明存在一个pent（4，$ r $），$ j $相对线对，所有足够大的可允许$ r $ $。使用一些具有块5的新组可分割设计（包括$ 2^{35} $，$ 2^{71} $和$ 10^{23} $），我们大大扩展了Pent的已知存在频谱（5，$ r $）。
A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line, or block, is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. An opposite line pair in a pentagonal geometry consists of two parallel lines such that each point on one of the lines is not collinear with precisely those points on the other line. We give a direct construction for an infinite sequence of pentagonal geometries with block size 3 and connected deficiency graphs. Also we present 39 new pentagonal geometries with block size 4 and five with block size 5, all with connected deficiency graphs. Consequentially we determine the existence spectrum up to a few possible exceptions for PENT(4, $r$) that do not contain opposite line pairs and for PENT(4, $r$) with one opposite line pair. More generally, given $j$ we show that there exists a PENT(4, $r$) with $j$ opposite line pairs for all sufficiently large admissible $r$. Using some new group divisible designs with block size 5 (including types $2^{35}$, $2^{71}$ and $10^{23}$) we significantly extend the known existence spectrum for PENT(5, $r$).
